Abstract : Context. The origin and life-cycle of molecular clouds are still poorly constrained, despite their importance for understanding the evolution of the interstellar medium. Many large-scale surveys of the Galactic plane have been conducted recently, allowing for rapid progress in this field. Nevertheless, a sub-arcminute resolution global view of the large-scale distribution of molecular gas, from the diffuse medium to dense clouds and clumps, and of their relationshipto the spiral structure, is still missing. Aims: We have carried out a systematic, homogeneous, spectroscopic survey of the inner Galactic plane, in order to complement the many continuum Galactic surveys available with crucial distance and gas-kinematic information. Our aim is to combine this data set with recent infrared to sub-millimetre surveys at similar angular resolutions. Methods: The SEDIGISM survey covers 78 deg2 of the inner Galaxy (-60°≤ℓ≤ 18°, |b|≤ 0.5°) in the J = 2-1 rotational transition of 13CO. This isotopologue of CO is less abundant than 12CO by factors up to 100. Therefore, its emission has low to moderate optical depths, and higher critical density, making it an ideal tracer of the cold, dense interstellar medium. The data have been observed with the SHFI single-pixel instrument at APEX. The observational setup covers the 13CO(2-1) and C18O(2-1) lines, plus several transitions from other molecules. Results: The observations have been completed. Data reduction is in progress, and the final data products will be made available in the near future. Here we give a detailed description of the survey and the dedicated data reduction pipeline. To illustrate the scientific potential of this survey, preliminary results based on a science demonstration field covering -20°≤ℓ ≤ -18.5° are presented. Analysis of the 13CO(2-1) data in this field reveals compact clumps, diffuse clouds, and filamentary structures at a range of heliocentric distances. By combining our data with data in the (1-0) transition of CO isotopologues from the ThrUMMS survey, we are able to compute a 3D realization of the excitation temperature and optical depth in the interstellar medium. Ultimately, this survey will provide a detailed, global view of the inner Galactic interstellar medium at an unprecedented angular resolution of 30''. This publication is based on data acquired with the Atacama Pathfinder EXperiment (APEX) under programmes 092.F-9315(A) and 193.C-0584(A). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.Full Table 5 and Table A.1 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/601/A124